# Heptagonal tiling honeycomb

Heptagonal tiling honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {7,3,3} |
| Coxeter diagram | |
| Cells | {7,3} |
| Faces | Heptagon {7} |
| Vertex figure | tetrahedron {3,3} |
| Dual | {3,3,7} |
| Coxeter group | [7,3,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the heptagonal tiling honeycomb or 7,3,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

## Geometry

The Schläfli symbol of the heptagonal tiling honeycomb is {7,3,3}, with three heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a tetrahedron, {3,3}.
| Poincaré disk model (vertex centered) | Rotating | Ideal surface |

## Related polytopes and honeycombs

It is a part of a series of regular polytopes and honeycombs with {p,3,3} Schläfli symbol, and tetrahedral vertex figures:
{p,3,3} honeycombs :
| Space | Space | S$^3$ | S$^3$ | S$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ |
| Form | Form | Finite | Finite | Finite | Paracompact | Noncompact | Noncompact | Noncompact |
| Name | Name | {3,3,3} | {4,3,3} | {5,3,3} | {6,3,3} | {7,3,3} | {8,3,3} | ... {∞,3,3} |
| Image | Image | | | | | | | |
| Coxeter diagrams | 1 | | | | | | | |
| Coxeter diagrams | 4 | | | | | | | |
| Coxeter diagrams | 6 | | | | | | | |
| Coxeter diagrams | 12 | | | | | | | |
| Coxeter diagrams | 24 | | | | | | | |
| Cells {p,3} | Cells {p,3} | {3,3} | {4,3} | {5,3} | {6,3} | {7,3} | {8,3} | {∞,3} |
It is a part of a series of regular honeycombs, {7,3,p}.
| {7,3,3} | {7,3,4} | {7,3,5} | {7,3,6} | {7,3,7} | {7,3,8} | ...{7,3,∞} |
It is a part of a series of regular honeycombs, with {7,p,3}.
| {7,3,3} | {7,4,3} | {7,5,3}... |

### Octagonal tiling honeycomb

Octagonal tiling honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {8,3,3} t{8,4,3} 2t{4,8,4} t{4$^{[3,3]}$} |
| Coxeter diagram | (all 4s) |
| Cells | {8,3} |
| Faces | Octagon {8} |
| Vertex figure | tetrahedron {3,3} |
| Dual | {3,3,8} |
| Coxeter group | [8,3,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the octagonal tiling honeycomb or 8,3,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an octagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the octagonal tiling honeycomb is {8,3,3}, with three octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an tetrahedron, {3,3}.
| Poincaré disk model (vertex centered) | Direct subgroups of [8,3,3] |

### Apeirogonal tiling honeycomb

Apeirogonal tiling honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {∞,3,3} t{∞,3,3} 2t{∞,∞,∞} t{∞$^{[3,3]}$} |
| Coxeter diagram | (all ∞) |
| Cells | {∞,3} |
| Faces | Apeirogon {∞} |
| Vertex figure | tetrahedron {3,3} |
| Dual | {3,3,∞} |
| Coxeter group | [∞,3,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the apeirogonal tiling honeycomb or ∞,3,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an apeirogonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,3,3}, with three apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an tetrahedron, {3,3}.
The "ideal surface" projection below is a plane-at-infinity, in the Poincare half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.
| Poincaré disk model (vertex centered) | Ideal surface |